I am wondering if the biometric information collected by Apple's Health app stays local on my iPhone, or if it's being sent to Apple?
Unless you choose to backup your data to iCloud, none of your data will be sent to Apple. Even if you do choose to backup your data to iCloud, that data is encrypted - meaning only you can access it.
To quote Apple directly:
In a nutshell, only you (or anyone who knows your passcode or has a fingerprint set on your iPhone) have control of your health data.